The present invention relates, in general, to cryptography, and, in particular, to electric signal modification (e.g., scrambling).
The Fourier transform is used to transform a signal in the time domain into a signal in the frequency domain. The fractional Fourier transform is used to transform a signal in the time domain to a signal in the frequency domain, but with a user-definable angle of rotation.
The fractional Fourier transform of a signal S(t) is defined as follows.
Fxcex1S(y)=∫S(t)Kxcex1(t)dt
The kernel of the fractional Fourier transform is as follows:
xe2x80x83Kxcex1(t, y)={square root over ((1xe2x88x92icot xcex1)/(2xcfx80))}exp{0.5i(y2+t2) cot xcex1xe2x88x92iytcsc xcex1}
if xcex1 is not an integer multiple of xcfx80, and
Kxcex1(t, y)=xcex4(txc2x1y)
if xcex1 is an integer multiple of xcfx80, where the sign of the argument in the delta distribution alternates with the parity of the integer, and where the variable i is the square root of xe2x88x921. Because the fractional Fourier transform kernel includes the square root of xe2x88x921, the kernel includes both a real component and an imaginary component.
The fractional Fourier transform is further described in an article entitled xe2x80x9cThe Fractional Fourier Transform and Time-Frequency Representations,xe2x80x9d by Luxc3xads B. Almeida, IEEE Transactions on Signal Processing, Vol. 42, No. 11, November 1994, pps. 3084-3091, and in an article entitled xe2x80x9cRelationships between the Radon-Wigner and fractional Fourier transforms,xe2x80x9d by Adolf W. Lohmann and Bernard H. Soffer, Journal of the Optical Society of America, Vol. 11, No. 6, June 1994, pps. 1798-1801. Neither article discloses the cryptographic method of the present invention.
U.S. Pat. No. 5,840,033, entitled xe2x80x9cMETHOD AND APPARATUS FOR ULTRASOUND IMAGING,xe2x80x9d uses the fractional Fourier transform as disclosed in the above-identified articles as an equivalent method of performing a two-dimensional Fourier transform. U.S. Pat. No. 5,840,033 does not disclose the cryptographic method of the present invention. U.S. Pat. No. 5,840,033 is hereby incorporated by reference into the specification of the present invention.
U.S. Pat. No. 5,845,241, entitled xe2x80x9cHIGH-ACCURACY, LOW-DISTORTION TIME-FREQUENCY ANALYSIS OF SIGNALS USING ROTATED-WINDOW SPECTROGRAMS,xe2x80x9d uses a fractional Fourier transform as disclosed in the above-identified articles to form rotated window spectrograms. U.S. Pat. No. 5,845,241 does not disclose the cryptographic method of the present invention. U.S. Pat. No. 5,845,241 is hereby incorporated by reference into the specification of the present invention.
It is an object of the present invention to encrypt and decrypt a signal using at least one component of a modified fractional Fourier transform kernel a user-definable number of times.
It is another object of the present invention to encrypt and decrypt a signal using at least one component of a modified fractional Fourier transform kernel a user-definable number of times with at least one encryption key and at least one decryption keys.
The present invention is a cryptographic method using at least one component of a modified fractional Fourier transform kernel a user-definable number of times. Cryptography encompasses both encryption and decryption.
The first step of the method of encryption is receiving a signal to be encrypted.
The second step of the method of encryption is establishing at least one encryption key, where each at least one encryption key includes at least four user-definable variables that represent an angle of rotation, a time exponent, a phase, and a sampling rate.
The third step of the method of encryption is selecting at least one component of a modified fractional Fourier transform kernel, where each at least one component of a modified fractional Fourier transform kernel selected corresponds to, and is defined by, one of the at least one encryption keys.
The fourth, and last, step of the method of encryption is multiplying the signal by the at least one component of a modified fractional Fourier transform kernel selected in the third step.
The first step of the method of decryption is receiving a signal to be decrypted.
The second step of the method of decryption is establishing at least one decryption key, where each at least one decryption key corresponds with, and is identical to, an encryption key used to encrypt the signal.
The third step of the method of decryption is selecting at least one component of a modified fractional Fourier transform kernel, where each at least one component of a modified fractional Fourier transform kernel selected corresponds with, and is identical to, a component of a modified fractional Fourier transform kernel used to encrypt the signal.
The fourth, and last, step of the method of decryption is dividing the signal by the at least one component of a modified fractional Fourier transform kernel selected in the third step.